Singular measure

In mathematics, two positive (or signed or complex) measures ${\displaystyle \mu }$ and ${\displaystyle \nu }$ defined on a measurable space ${\displaystyle (\Omega ,\Sigma )}$ are called singular if there exist two disjoint measurable sets ${\displaystyle A,B\in \Sigma }$ whose union is ${\displaystyle \Omega }$ such that ${\displaystyle \mu }$ is zero on all measurable subsets of ${\displaystyle B}$ while ${\displaystyle \nu }$ is zero on all measurable subsets of ${\displaystyle A.}$ This is denoted by ${\displaystyle \mu \perp \nu .}$

A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure.

As a particular case, a measure defined on the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is called singular, if it is singular with respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure.

Example. A discrete measure.

The Heaviside step function on the real line, $${\displaystyle H(x)\ {\stackrel {\mathrm {def} }{=}}{\begin{cases}0,&x<0;\\1,&x\geq 0;\end{cases}}}$$ has the Dirac delta distribution ${\displaystyle \delta _{0}}$ as its distributional derivative. This is a measure on the real line, a "point mass" at ${\displaystyle 0.}$ However, the Dirac measure ${\displaystyle \delta _{0}}$ is not absolutely continuous with respect to Lebesgue measure ${\displaystyle \lambda ,}$ nor is ${\displaystyle \lambda }$ absolutely continuous with respect to ${\displaystyle \delta _{0}:}$ ${\displaystyle \lambda (\{0\})=0}$ but ${\displaystyle \delta _{0}(\{0\})=1;}$ if ${\displaystyle U}$ is any non-empty open set not containing 0, then ${\displaystyle \lambda (U)>0}$ but ${\displaystyle \delta _{0}(U)=0.}$

Example. A singular continuous measure.

The Cantor distribution has a cumulative distribution function that is continuous but not absolutely continuous, and indeed its absolutely continuous part is zero: it is singular continuous.

Example. A singular continuous measure on ${\displaystyle \mathbb {R} ^{2}.}$

The upper and lower Fréchet–Hoeffding bounds are singular distributions in two dimensions.